Applied Game Theory Lecture 1

Applied Game TheoryLecture 1 Pietro Michiardi

Before we even start The Grade Game Without showing your neighbors what you are doing, write down on a form either the letter alpha or the letter beta. Think of this as a “grade bid”. I will randomly pair your form with one other form. Neither you nor your pair will ever know with whom you were paired. Here is how grades may be assigned for this class: • If you put alpha and your pair puts beta, then you will get grade A, and your pair grade C; • If both you and your pair put alpha, then you both will get the grade B-; • If you put beta and your pair puts alpha, then you will get the grade C and your pair grade A; • If both you and your pair put beta, then you will both get grade B+

What is game theory? • Game theory is a method of studying strategic situations • Informally: • At one end we have Firms in perfect competition: in this case, firms are price takers and do not care about what other do • At the other end we have Monopolist Firms: in this case, a firm doesn’t have competitors to worry about, they’re not price-takers but they take the demand curve • Everything in between is strategic, i.e., everything that constitutes imperfect competition • Example: The automotive industry

Literally, a strategic setting is one where the outcomes that affect you depend on actions of others, not only yours

The grade game, explained (1) • Just reading the text is hard to absorb, let’s use a concise way of representing the game: my pair my pair alpha beta alpha beta alpha B - A alpha B - C me me C B + A B + beta beta my grades pair’s grades

The grade game: outcome matrix • Let’s use some more compact representation: my pair alpha beta This is an outcome matrix: It tells us everything that was in the game we saw alpha B - , B - A , C me C , A B + , B + beta 1st grade Row player 2nd grade Column player

The grade game, explained (2) What did you do? • How many chose alpha? • How many chose beta? • Why?

The grade game, explained (3) • Regardless of my partner choice, there would be better outcomes for me by choosing alpha rather than beta; • We could all be collusive and work together, hence by choosing beta we would get higher grades. • What we have examined is not a game yet

The grade game, explained (4) • Right now we have: • The players • Strategies, that is the actions players can take • We know what the outcomes are • We are missing objectives, i.e. payoffs • Basically we don’t know what players care about

The grade game, explained (5) • In our previous discussion we had two different payoffs: • We had the ones where the only thing we care about was our own grade • We had the ones where we might care about other people’s payoff as well • We’re going to explore all possible combinations of payoffs in the next couple of slides

The grade game: payoff matrix • Possible payoffs: in this case we only care about our own grades my pair alpha beta # of utiles, or utility: (A,C)  3 (B-, B-)  0 Hence the preference order is: A > B+ > B- > C alpha 0 , 0 3, -1 me -1, 3 1,1 beta

The grade game, explained (6) • What do we call people who only care about their own grades? • So, now, given the payoff matrix, what should you do? • Play alpha! Indeed, no matter what the pair does, by playing alpha you would obtain a higher payoff Definition:We say that my strategy alpha strictly dominates my strategy beta, if my payoff from alpha is strictly greater than that from beta, regardless of what others do.

Lesson 1 Do not play strictly dominated strategies

Dominated strategies (1) • Why shouldn’t you play strictly dominated strategies? • Because if I play a dominating strategy I’m doing better than what I could do regardless what the other does • Let’s look again at the payoff matrix • If we (me and my pair) reason selfishly, we will both select alpha, and get a payoff of 0; • But if we reasoned in a different way, we could end up both with a payoff of 1;

Dominated strategies (2) • What’s the problem with this latter reasoning? • Suppose you have super mental power and oblige your partner to agree with you and chose beta, so that you both would end up with a payoff of 1… • Even with communication, it wouldn’t work, because at this point, you’d be better of by choosing alpha, and get a payoff of 3

Lesson 2 Rational choice (i.e., not choosing a dominated strategy) can lead to outcomes that suck!

The Prisoners’ Dilemma • Did you know it? • Any other examples? • What kind of remedies we have for such situations?

The grade game: payoff matrix • Possible payoffs: This time people are more incline to be altruistic my pair alpha beta # of utiles, or utility: (A,C)  3 – 4 = -1 my ‘A’ my guilt (C, A)  -1 – 2 = -3 my ‘C’ my indignation This is a coordination problem alpha 0 , 0 -1, -3 me -3,-1 1,1 beta

The grade game, explained (6) • What would you do in this case? • By choosing alpha you may “minimize your losses” • By choosing beta you may “maximize your profit” • We have the same game structure, the same outcomes, but the payoffs are different • Is there any dominated strategy in this game?

Lesson 3: Payoffs matters! << … you can’t get what you want unless you know what you want … >>

The grade game: payoff matrix • Selfish vs. Altruistic my pair (Altruistic) alpha beta In this case, alpha still dominates The fact we (selfish player) are playing against an altruistic player doesn’t change my strategy, even by changing the other Player’s payoff alpha 0 , 0 3, -3 Me (Selfish) -1,-1 1,1 beta

The grade game: payoff matrix • Altruistic vs. Selfish my pair (Selfish) alpha beta • What happened here? • Do I have a dominating strategy? • Does the other player have a dominating • strategy? • By thinking of what my “opponent” will do • I can decide what to do. alpha 0 , 0 -1, -1 Me (Altruistic) -3,3 1,1 beta

Lesson 4: Put yourself in others’ shoes and try to figure out what they will do

Observations • In realistic settings: • It is often hard to determine what are the payoffs of your “opponent” • It is easier to figure out my own payoffs • In general, we have to figure out what are the odds (probability) of my “opponent” being selfish or altruistic

A slightly more complicated game The “Pick a Number” Game Without showing your neighbor what you’re doing, write down an integer number between 1 and 100. I will calculate the average number chosen in the class. The winner in this game is the person whose number is closest to two-thirds of the average in the class. The winner will win 5 euro minus the difference in cents between her choice and that two-thirds of the average. Example: 3 students Numbers: 25, 5, 60 Total: 90, Average: 30, 2/3*average: 20 25 wins: 5 euro – 5cents = 4.95 euro

While we sum-up in the next couple of slides, pick your number, write it down and give it to me…

The story so far… (1) • We’ve seen a compact representation of games: this is called the normal form • Lessons we learned: • Do not play strictly dominated strategies • Put yourself in others’ shoes • It doesn’t just matter what your payoffs are • It’s also important what other people’s payoff are, because you want to try and figure out what they’re going to do and respond appropriately

The story so far … (2) • We’ve seen an important class of games: Prisoners’ Dilemma games • Joint project: • Each individual may have an incentive to shirk • Price competition • Each firm has an incentive to undercut prices • If all firms behave this way, prices are driven down towards marginal cost and industry profit will suffer • Common resource • Carbon emissions • Fishing

The story so far… (3) • In each of the previous examples we end up with a bad outcome • This is not a failure of communication • Solutions: • Contracts  change the payoffs • Repeated interaction

Let me introduce some notationand by the way… hand in your numbers

Assumptions • We assume all the ingredients of the game to be known • Everybody knows the possible strategies everyone else could choose • Everybody knows everyone else’s payoffs • This is not very realistic, but things are complicated enough to give us material for this class

Example 1 2 C L R T 1 B NOTE: This game is not symmetric

Example… continued • How is the game going to be played? • Does player 1 have a dominated strategy? • Does player 2 have a dominated strategy? • For a strategy to be dominated, we need another strategy for the same player that does always better (in terms of payoffs)

Example 2: “Hannibal” game • An invader is thinking about invading a country, and there are 2 ways through which he can lead his army. • You are the defender of this country and you have to decide which of these ways you choose to defend: you can only defend one of these routes. • One route is a hard pass: if the invader chooses this route he will lose one battalion of his army (over the mountains). • If the invader meets your army, whatever route he chooses, he will lose a battalion

Example 2: “Hannibal” game attacker e, E = easy ; h,H = hard Payoffs are how many battalions attacker will arrive with in your country h e E defender H

Example 2: “Hannibal” game • You’re the defender: what would you do? • Is it true that defending the easy route dominates defending the hard one? • You’re the attacker: what would you do? • Now, what the defender should do, if he would put himself in the attacker shoes?

Back to the “pick a number” game • What we know: • Do not chose a strictly dominated strategy • Also, do not chose a weakly dominated strategy • You should put yourself in others’ shoes, try to figure out what they are going to play, and respond appropriately • What did you do in this game?

Back to the “pick a number” game • A possible assumption: • People chose numbers uniformly at random • The average is 50 • 2/3 * average = 33.3 • What’s wrong with this reasoning?

Back to the “pick a number” game • Let’s try to find out whether there are dominated strategies • If everyone would chose 100, then the winning number would be 66 • numbers > 67 are weakly dominated by 66 • Rationality tells not to choose numbers > 67

Back to the “pick a number” game • So now we’ve eliminated dominated strategies, it’s like the game was to be played over the set [1, …, 67] • Once you figured out that nobody is going to chose a number above 67, the conclusion is • Also strategies above 45 are ruled out • They are weakly dominated, only once we delete 68-100 • This implies rationality, and knowledge that others are rational as well

Back to the “pick a number” game • Eventually, we can show that also strategies above 30 are weakly dominated, once we delete previously dominated strategies • We can go on with this line of reasoning and end up with the conclusion that: • 1 was the winning strategy!!

Back to the “pick a number” game • Common knowledge: you know that others know that others know … and so on that rationality is underlying all players’ choices • In practice: • Average number was: • Winning number was: 2/3*average

Theory vs. Practice • Why was it that 1 wasn’t the winning answer? • We need a strong assumption, that is that all players are rational and they know that everybody else’s rational as well

To sum up … (1) • We’ve explored a bit the idea of deleting dominated strategies • Look at a game • Figure out which strategies are dominated • Delete them • Look at the game again • Look at which strategies are dominated now • … and so on …

To sum up … (2) • Iterative deletion of dominated strategies seems a powerful idea, but it’s also dangerous if you take it literally • In some games, iterative deletion converges to a single choice, in others it may not (we’ll see shortly an example) • HINT: try to identify all dominated strategies at once before you delete, this may save you some rounds…

Our first model: politics (1) • Imagine there are 2 candidates • These candidates are choosing their political positions on a spectrum • To make life easy let’s assume the spectrum has 10 positions 1 2 3 4 5 6 7 8 9 10 LEFT WING RIGHT WING

Our first model: politics (2) • We assume that there are 10% of the voters at each of these positions: • Voters are uniformly distributed • We assume voters will eventually vote for the closest candidate, that is for the candidate whose position is closest to their own • We break ties by splitting votes equally • We have players (candidates), and actions (political positions): what are we missing?

Applied Game Theory Lecture 1